Page:Elementary lectures on electric discharges, waves and impulses, and other transients (Steinmetz 1911).djvu/27

To produce the magnetic field $\Phi$ of the current $i$, a voltage $e'$ must be consumed in the circuit, which with the current $i$ gives the power $p$, which supplies the stored energy $w$ of the magnetic field $\Phi$. This voltage $e'$ is called the inductance voltage, or voltage consumed by self-induction.

Since no power is required to maintain the field, but power is required to produce it, the inductance voltage must be propor- tional to the increase of the magnetic field:

(3)

$e'={\frac {d\Phi }{dt}}$,

or by (1),

(4)

$e'=L{\frac {di}{dt}}$.

If $i$ and therefore $\Phi$ decrease, ${\frac {di}{dt}}$ and therefore $e'$ are negative; that is, $p$ becomes negative, and power is returned into the circuit.

The energy supplied by the power $p$ is

$w=\int {}{}p\,dt$,

or by (2) and (4),

$w=\int {}{}Li\,di$;

hence

(5)

$w={\frac {Li^{2}}{2}}$

is the energy of the magnetic field

$\Phi =Li$

of the circuit.

9. Exactly analogous relations exist in the dielectric field.

The dielectric field, or dielectric flux, ${\mathit {\Psi }}$, is proportional to the voltage $e$, with a proportionality factor, $C$, which is called the capacity of the circuit:

(6)

$\Psi =Ce$.

The dielectric field represents stored energy, $w$. To produce it, power, $p$, must, therefore, be supplied by the circuit. Since power is current times voltage,

(7)

$p=i'e$.

To produce the dielectric field ${\mathit {\Psi }}$ of the voltage $e$, a current $i'$ must be consumed in the circuit, which with the voltage $e$ gives